Rational root theorem problem 
The polynomial $f(x)=15x^3-86x^2-28x+24$ has three rational roots. What is the largest positive difference between any two of them?

How should one approach this? Using the rational root theorem I end up with lots of possible candidates and it seems that I'm not approaching this the right way. Can we use the information that they give us about $f(x)$ having three rational roots?
 A: Here's a solution that uses the assumption that the roots are all rational and avoids doing much messy arithmetic.
Since $f(-1)=-15-86+28+24$ is clearly negative, $f(0)=24$ is positive, and $f(1)=15-86-28+24$ is clearly negative again, one of the roots is between $-1$ and $0$, another is between $0$ and $1$, and the third is greater than $1$. Consequently, if the roots are all rational, then $f(x)$ must factor into the form $(x-a)(3x-b)(5x-c)$ where $a$ is an integer greater than $1$, $abc=-24$, and $0\lt|b/3|,|c/5|\lt1$, with one of $b$ and $c$ positive and the other negative.
Now writing $f(a)=((15a-86)a-28)a+24$, it's easy to see that we must have $a\gt5$, which limits it at worst to the values $6$, $8$, $12$, and $24$. But it's also not hard to see that 
$$f(6)=((90-86)6-28)6+24=(4\cdot6-28)6+24=-4\cdot6+24=0$$
so we've identified the integer root. This leads to the factorization
$$15x^3-86x^2-28x+24=(x-6)(15x^2+4x-4)=(x-6)(3x+2)(5x-2)$$
which gives the other two roots, namely $-2/3$ and $2/5$. We now see that the largest positive difference between roots is
$$6-(-2/3)=20/3$$
A: Render $x=2y$, then
$15y^3-43y^2-7y+3=0$
which, with the smaller constant coefficient, will require fewer trials.
The sum of the roots for $y$ is $43/15$ so their average is $43/45$. If all three roots are to be rational and have this average then at least one of them must be $1$ or $3$, as these are the only candidates greater than or equal to the average.  Clearly $y=1$ won't work, so $y=3$ must work and this corresponds to $x=2y=6$ in the original equation.  You can then factor this out and solve the remaining quadratic equation.
